The isomorphism problem for groups is to determine whether two finite groups are isomorphic. The groups are assumed to be epresented by their multiplication tables. Tarjan has shown that this problem can bedone in time O(nlog n+O(1)) for groups of cardinality n. Savage has claimed an algorithm for showing that if the groups are Abelian,isomorphism can be determined in time O(n2). We improve this bound and give an O(n log n) algorithm for Abelian group isomorphism.Further, we show that if the groups are Abelian p-groups then isomorphism can be determined in time O(n). We also show that the elementary divisor sequence for an Abelian group can be determined in time O(n log n) and for an Abelian p-group it can be determined in time O(n).